Split a unit fraction among a whole number of people and keep each share tied to the original whole.
Listen for the idea
A unit fraction divided by a positive whole number gives a smaller equal share of the original whole.
Represent 1/b ÷ n as n equal shares of one b-th.
Name each share relative to the original whole and verify by multiplication.
Materials: Paper rectangles; Pencil and ruler.
A 15-minute route
3 minutes: Try the three diagnostic questions. Ask for a brief explanation.
3 minutes: Teach one worked example that addresses the difficulty.
6 minutes: Choose three practice tasks and discuss the reasoning.
3 minutes: Give one transfer task independently and decide the next step.
The second worked example, three unused practice tasks, and remaining transfer task are available for a follow-up session. Allow more time if the group needs to build the models.
Teaching move: Outline the original whole, shade only the amount available, then split that shaded amount among the recipients. Repeat the subdivisions across the original whole to name one share.
Use the answer notes to interpret what you hear. Choose a response for each question to get a suggested teaching step. These choices describe this brief check and are not a mastery score.
1 One half of a sandwich is shared equally by 3 people. Does each person receive more or less than half a sandwich?
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: Less than half; each receives 1/6 of the original whole sandwich.
Listen for this difficulty: Interpreting division as counting how many halves fit in three sandwiches.
Try this response: Keep only one half available and separate it into three equal shares.
Look for:
Predicts less than 1/2.
Names 1/6 of the original sandwich.
2 A student writes 1/4 ÷ 2 = 2/4. Explain why this cannot describe equal sharing.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: Sharing the available fourth between two people gives each 1/8, not 2/4; two fourths exceed the entire available amount.
Listen for this difficulty: Multiplying the unit fraction by the number of recipients.
Try this response: Check whether two copies of the claimed share fit into the original fourth.
Look for:
Rejects 2/4 by checking the available amount.
Finds 1/8.
3 One third of a strip is shared among 4 people. A student calls each share 1/4. Of which whole is that true, and what fraction of the original strip is each share?
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: Each share is 1/4 of the available third, but 1/12 of the original whole strip.
Listen for this difficulty: Changing the reference whole from the original strip to the available amount.
Try this response: Keep two labeled outlines: the full strip and its available third.
Look for:
Distinguishes the two reference amounts.
Names 1/12 of the original strip.
2 · Model together
Two worked examples
Use one now. Ask the learner to explain each step before revealing the next part of your drawing.
1 Share 1/3 meter of ribbon equally among 4 people. How much does each receive?
Draw one meter and divide it into three equal parts.
Only one third is available; split it into four equal shares.
Imagine the same four-way split in each third: twelve equal pieces fill one meter.
Each share is 1/12 meter; four shares make 4/12 = 1/3 meter.
Result: 1/12 meter per person.
Teaching notes
Listen for this difficulty: Answering one fourth without keeping the original meter as the unit.
Try this response: Point to the one-meter outline whenever naming a fraction of a meter.
Look for:
Finds 1/12 meter.
Checks four equal shares reconstruct 1/3 meter.
2 One fifth of a pan of cornbread is shared equally between 2 people. Write an equation and a multiplication check.
Label the full pan as one whole and identify one fifth.
Cut that fifth into two equal pieces.
Each fifth could hold two such pieces, so ten fit in the original pan.
Give one piece to each person and check the two tenths equal one fifth.
Result: 1/5 ÷ 2 = 1/10 pan; 2 × 1/10 = 1/5.
Teaching notes
Listen for this difficulty: Dividing the denominator by two or giving each person one fifth.
Try this response: Track the total distributed amount as two copies of the proposed share.
Look for:
Writes 1/10 of a pan per person.
Checks with multiplication by 2.
3 · Guided practice
Choose three tasks
Mix a calculation or drawing with an explanation or error repair. Keep the other tasks for another short session.
1 Share 1/2 liter of juice equally into 4 cups. How much goes in each cup?
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 1/8 liter.
Listen for this difficulty: Treating four cups as four whole liters to divide by a half.
Try this response: Draw only the half liter available and split it into four equal portions.
Look for:
Finds 1/8 liter per cup.
Checks four portions make 1/2 liter.
2 Find 1/6 ÷ 3 and explain how many equal shares of that size fit in the original whole.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 1/18; eighteen such shares fit in one whole.
Listen for this difficulty: Using 6 + 3 for the new denominator.
Try this response: Split each of six sixths into three equal pieces.
Look for:
Finds 1/18.
Connects 6 × 3 to pieces per whole.
3 A student shares 1/4 of a pie among 3 people and gives each 1/12 of the pie. Verify the answer without repeating a division rule.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: Three shares total 3 × 1/12 = 3/12 = 1/4, so the answer is correct.
Listen for this difficulty: Accepting a fraction without checking it reconstructs the starting amount.
Try this response: Join three twelfth-sized pieces and compare them with the available fourth.
Look for:
Uses multiplication or recombination.
Shows the total is exactly the available fourth.
4 Which gives a larger share: 1/3 ÷ 2 or 1/3 ÷ 5? Explain before calculating.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 1/3 ÷ 2 is larger. The same third shared by fewer people gives larger shares: 1/6 versus 1/15.
Listen for this difficulty: Thinking a larger divisor gives a larger share.
Try this response: Keep the available third fixed and compare two-way with five-way partitions.
Look for:
Chooses the two-person share.
Explains 1/6 > 1/15 using equal available amounts.
5 One eighth of a cake is given to one person. Write the division and result.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 1/8 ÷ 1 = 1/8.
Listen for this difficulty: Assuming every division must make an amount strictly smaller.
Try this response: Ask whether sharing with one recipient splits the available piece at all.
Look for:
Finds 1/8.
Explains that dividing by 1 keeps the amount.
6 Make a story for 1/5 ÷ 3. Include the available amount, number of equal shares, and amount in one share.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: Example: 1/5 liter is poured equally into 3 cups, with 1/15 liter in each.
Listen for this difficulty: Writing an integer-divided-by-unit-fraction story for the reversed operation.
Try this response: Start the story with exactly one fifth available, not five or three whole units.
Look for:
Matches a unit-fraction available amount and 3 recipients.
Finds 1/15 in the same original unit; equivalent stories count.
4 · Independent transfer
Try a fresh situation
Choose one task without coaching. Check the explanation against both criteria in its teaching notes; use the other task for a later check.
1 One quarter kilogram of clay is shared equally among 6 students. Explain the amount per student with a diagram description and a multiplication check.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 1/24 kilogram per student. Split each quarter of a kilogram into six pieces; each piece is one twenty-fourth. Six pieces give 6/24 = 1/4 kilogram.
Listen for this difficulty: Calling each share one sixth of a kilogram instead of one sixth of the available quarter.
Try this response: Keep the original kilogram outline visible around all four quarters.
Look for:
Finds 1/24 kilogram.
Explains the whole and verifies six shares reconstruct 1/4 kilogram.
2 A group has 1/3 liter of paint. Three people will share it equally. Another group has 3 liters and fills 1/3-liter jars. Explain which situation is 1/3 ÷ 3 and solve only that one.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: The equal-sharing group is 1/3 ÷ 3 = 1/9 liter per person. The jar situation asks how many thirds fit in three liters.
Listen for this difficulty: Confusing the available amount with the size of each group.
Try this response: Underline what is known in each story: number of recipients versus size of each jar.
Look for:
Selects the equal-sharing situation.
Finds 1/9 liter per person and distinguishes the meanings.
What to do next
If the explanation holds: Continue when the student names each share in the original unit and verifies that all recipients together use exactly the available amount.
If the difficulty remains: Return to half a paper square shared by two people. Keep the full square outline and compare a share with the original whole.
Standards: 5.NF.B.7.a, 5.NF.B.7.c. Grade 5 alignment describes the mathematical work in this pack.
This division work is limited to a unit fraction divided by a positive whole number and a positive whole number divided by a unit fraction, as specified in Grade 5 NF.7.
The tasks, worked examples, and teacher responses are original Fraction Classroom materials, version 2.0.0. The teaching approach draws on noticing the unit, moving between representations, and responding to student reasoning in the linked Illustrative Mathematics lesson. Its text and exercises are not reproduced here.